Two conditional expectations equal almost everywhere

Question

Suppose $X$ is a continuous random variable. If $mathbb{E}[X,|,mathcal{F}]=mathbb{E}[X,|,mathcal{G}]$ almost everywhere for two sub-sigma algebra $mathcal{F}$ and $mathcal{G}$, does this imply $mathcal{F}$ and $mathcal{G}$ are set theoretically identical?

Kavi Rama Murthy · Answer

No. If $X,Y,Z$ are independent, $mathcal F=sigma (Y)$ and $mathcal G =sigma (Z)$ then $mathbb E( X|mathcal F)=mathbb E(X|mathcal G)=mathbb EX$.

Two conditional expectations equal almost everywhere

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